Several relaxed galaxy clusters show the presence inside their Intra Cluster Medium (ICM) of couples of cavities in the X-ray emission, usually located in opposite directions compared to the central Brightest Cluster Galaxy [1]}, [2]}. The origin of these cavities is probably due to the mechanical power produced by AGN jets that is converted to heat during the expansion of the lobe in the ICM [3]}. Many X-ray cavities host a diffuse radio emission [4]}, showing that relativistic electrons and magnetic fields are present in them. However, it has been found that the energy stored in these relativistic electrons, determined under the minimum energy assumption, is well below the energy necessary to inflate the cavity [5]}, indicating that another component inside the cavity should be dominant in energy. This dominant component can be a high temperature thermal gas with a temperature of the order of \(kT\sim 100\)  keV or higher according to hydrodynamic simulation [6]}, [7]}, or a population of relativistic protons.
